Problem 2 (30 points): Household Labor Supply Consider a household with the same utility function as in Problem 1. The household has a total time endowment of h = 1 and is paid a wage w for every unit of labor supplied. The household receives zero non-labor income (i.e. dividends less lump-sum taxes is equal to zero) but pays a labor income tax of 'r for every dollar earned. a Write down the household's problem. In your answer, state what is the household's objec- tive and what constraints it may face. b Solve for optimal consumption and leisure of the household. c Suppose all households are identical, and there are M households in the economy. What is aggregate consumption spending and aggregate labor supply equal to? d Suppose the tax rate 7' rises. What happens to aggregate consumption and aggregate labor supplied? Given your results, suggests what this implies about the relative magnitudes of the income and substitution effects. e Now, suppose that w = 2. If the government wants to maximize its revenue, what should be the optimal level of marginal tax rate, 7'? Problem 1 (10 points): Working with the Utility Function This question is to get you started. Consider the household with the following utility function: U(c, 6') : log(c) + 2 log(). a Derive expressions for the marginal utility of leisure and the marginal utility of consump- tion. b Derive an expression for the marginal rate of substitution between leisure and consump- tion, M RSgac